$\text{ Area of triangle formed by common tangents to the circle }$$x^2+y^2-6x=0\text{ and }{x}^2+y^2+2x=0\text{ is}$

$\text{ A divides }{\mathrm{C}}_1{\mathrm{C}}_2\text{ externally in the ratio }1:3\text{ . }$

$\begin{array}{l}\text{The external center of similitude is }P,{\text{\hspace{0.17em} it divides C}}_1{C}_2\text{\hspace{0.17em} in the ratio }{r}_1:r_2\text{\hspace{0.17em}externally. }\\ \text{externally. }\\ \text{Hence, }P\left(-3,0\right)\\ \text{Suppose that the angle between the direct common tangent is }\theta \\ \text{tan}\frac{\theta }{2}=\frac{r}{\sqrt{S_{11}}}=\frac{1}{\sqrt{3}}\\ \frac{\theta }{2}=30°\\ \text{Equation of tangent from }A\left(-3,0\right)\text{\hspace{0.17em} is a line making an angle 30}°\text{\hspace{0.17em} with }x-\text{\hspace{0.17em}axis }\\ \text{and passing through the point }\left(-3,0\right)\\ \text{Equation of tangent is }x+3=\sqrt{3}y\\ \text{it cuts the transverse common tangent }x=0\text{\hspace{0.17em} at }\left(0,\sqrt{3}\right)\\ \text{Therefore, the vertices of the required triangle are }\left(0,\sqrt{3}\right).\left(0,-\sqrt{3}\right),\left(-3,0\right)\\ \text{Hence the area of the triangle is 3}\sqrt{3}\text{\hspace{0.17em}square units}\end{array}$